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tl;dr: It depends on the ding. Try some cleaner wax, though: it won't make the problem worse.

Did you scratch down to the primer or just leave some paint behind?

If all you did was trade some paint between cars, you're going to need to polish it off. That said, if you're not comfortable with power polishing tools or abrasive polishes, I would counsel a cautious approach and shoot for incremental improvements.

CAVEAT: never wax or buff in direct sunlight. Misery awaits. Once upon a time, I put a whole set of swirl marks into my paint trying to get over-dried wax off in the sun.

So, here's what I used to do (before I bought the above-mentioned power tools):

Purchase some cleaner wax AKA a combination of car wax and mild abrasives (e.g., Zymol which works pretty well and smells nice in the process...)

Wash the area with nice sudsy dish soap. This will strip off any waxes that are currently on that body panel. You want easy access to the accidental paint.

No, seriously: wash the panel really well. You don't have to do the whole car, just the panel of interest.

Dry it well: again, you're trying to prevent any barriers between your upcoming efforts and the bad paint.

Apply the cleaner wax to a target area according to instructions. Don't get aggressive at this point: you're trying to lift a little of the mark, not scratch or haze the surrounding area. I usually use an all-cotton cloth with cross-hatch motions with a bias towards vertical passes on side panels.

Again, following instructions, buff the area when the wax dries / gets hazy. See above note for a reminder that impatience and / or direct sunlight can add to your misery factor.

If all is well, rewax the entire door panel that you stripped with the dish soap above. Nice cleaner wax will do a swell job for the larger panel as well.

Show the tolerant spouse the fruits of your labors. Receive accolades.

The cleaner wax is doing a few things: it's abrading some of the mark loose with each application, suspending those loose particles in the wax and carrying them away when you buff the dry wax off. In addition, the remaining buffed wax softens the edges of any dent or scratch that you made in the wife's car (i.e., they look less bad). Finally, the optical properties of the wax will reduce the probability that a glancing light will catch a perfect specular reflection (i.e., remaining scratches are harder to see).

Note: I have a random orbital at home and I go straight to mild polishes in these situations. This is an investment and not a step that I'd recommend to a novice or those in a hurry to get some immediate gratification.

Awesome. Not sure if this warrants a different question, but when you refer to 'buffing', what is the proper way to buff? To me it sounds intuitive, 'Karate Kid' wax on, wax off, so is it just taking a dry cloth and literally rubbing the wax off?
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BigHomieFeb 12 '14 at 13:58

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@MDMoore313, the product will often have useful directions. I usually wait until the wax is dry / hazy enough to see clearly. Use of cotton cloths will avoid scratchy polyester fibers. I tend to bias towards vertical passes on the sides of the car and fore-aft passes on the horizontal surfaces.
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Bob Cross♦Feb 12 '14 at 14:08